Convergence of the method of integral equations for quasi three-dimensional problem of electrical sounding

Method of Integral Equations for the Problem of Electrical Tomography in a Medium with Ground Surface Relief

“…Compared with a case when the surface is plain, the presence of ground surface relief makes a problem formulation for the method of integral equations more complicated. This is because we can use the method of reflections at a plain surface of discontinuity at the ground surface, which allows us to use a relatively simple system of integral equations [23,24] (in the case of multiple internal contacts of boundaries) [22]. To calculate the field at the ground surface, we define a distribution of current densities at the surface, which compensates for an external source to satisfy the physical boundary condition without current normal to the ground surface boundary:…”

“…The main new feature of the presented work is that we apply the method of integral equations for calculation of fields in a medium with a relief surface for the first time. Different applications of the integral equations method in geophysics are considered in articles [17][18][19][20][21][22][23][24] and are well established among the methods of mapping electrical resistivity.…”

Simulation of the Electric Field of a Point Source on the Relief Surface

“…Today we know that mostly often used method of solving direct problems is finite differences and finite element methods. They are useful for a big amount of practical aims, but they have several disadvantages [5]. In particular, these are: low accuracy of field's first derivatives computation, and subsequently computations of apparent resistivity, problem of creating the grid, adapted to complicated form of the boundary of a computation domain, problem of producing of artificial boundary conditions on boundaries of computation domain, problem of singularity, related to the function of the source and to the form of the relief.…”

“…Obviously, all the circuits of the cell which have a rectangular shape, comprise geometric error in consideration of the effect of the relief. It is possible to reduce the error only by increasing the number of cells, which leads to a significant increase in computational cost [5].…”